## Monday, March 16, 2015

### 2015 Mathcounts State Prep: Simon's Favorite Factoring Trick

Check out Mathcounts here, the best competition math program for middle school students.

The most common cases of Simon's Favorite Factoring Trick are:

I:  $$xy+x+y+1=\left( x+1\right) \left( y+1\right)$$

II:  $$xy-x-y+1=\left( x-1\right) \left( y-1\right)$$

It's easy to learn. Here is the best tutorial online, by none other than Richard Rusczyk.
The method Rusczyk uses at the second half is very nifty. Thanks!!

Questions to ponder:(answer key below)
#1: Both x and y are positive integers and $$x>y$$. Find all positive integer(s) that $$xy+x+y=13$$
#2: Both x and y are positive integers and $$x>y$$. Find all positive integer(s) that $$2xy+2x-3y=18$$
#3: Find the length and the width of a rectangle whose area is equal to its perimeter.
#4: Twice the area of a non-square rectangle equals triple it's perimeter, what is the area of the rectangle?

#1:  x = 6 and y = 1
#2: ( x, y ) = (4, 2)
#3: Don't forget square is a kind of rectangle (but not the other way around) so there are two answers:
4 by 4 and 3 by 6 units.
#4: One side is 4 units and the other 12 units so the answer is 4 x 12 or 48 square units.
There is another one, 6 by 6 that would fit if the question doesn't specify non-square rectangle.